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  "body": "\n## Definition\n\nLet $(M, \\omega)$ be a [[symplectic manifold|CAN490EPRT]] and let $H \\colon M \\to \\R$ be a smooth function.\nThe *Hamiltonian vector field* of $H$ is the unique [[vector field|0299VCW1TH]] $X_H$ on $M$ with\n$$\\iota_{X_H} \\omega = dH.$$\nThat is, $\\omega(X_H, Y) = dH(Y)$ for every tangent vector $Y$ at every point.\nA vector field is called *Hamiltonian* if it equals $X_H$ for some smooth $H$.\n\n## Remarks\n\nThe vector field exists and is unique because $\\omega$ is nondegenerate, so $v \\mapsto \\iota_v \\omega$ is an isomorphism from each tangent space onto the cotangent space.\n\nExample: on $\\R^{2n}$ with coordinates $(q_1, \\ldots, q_n, p_1, \\ldots, p_n)$ and $\\omega = \\sum_i dq_i \\wedge dp_i$, set\n$$X_H = \\sum_i \\left( \\frac{\\partial H}{\\partial p_i} \\frac{\\partial}{\\partial q_i} - \\frac{\\partial H}{\\partial q_i} \\frac{\\partial}{\\partial p_i} \\right).$$\nThen $\\iota_{X_H} \\omega = \\sum_i \\left( dq_i(X_H)\\, dp_i - dp_i(X_H)\\, dq_i \\right) = \\sum_i \\left( \\frac{\\partial H}{\\partial p_i} dp_i + \\frac{\\partial H}{\\partial q_i} dq_i \\right) = dH$.\nSo the integral curves of $X_H$ solve Hamilton's equations $q_i' = \\partial H / \\partial p_i$, $p_i' = -\\partial H / \\partial q_i$.\n\nNon-example: on $\\R^4$ the form $\\omega = dq_1 \\wedge dp_1$ is closed but degenerate.\nFor $H = q_2$ there is no vector field $X$ with $\\iota_X \\omega = dq_2$, because $\\iota_X \\omega$ is always a combination of $dq_1$ and $dp_1$.\n\nSome authors use the opposite sign, $\\iota_{X_H} \\omega = -dH$, together with $\\omega = \\sum_i dp_i \\wedge dq_i$; this gives the same vector field on $\\R^{2n}$.\nThe function $H$ is constant along the flow of $X_H$, since $dH(X_H) = \\omega(X_H, X_H) = 0$.\n",
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